Periodic Reeb flows and products in symplectic homology

TL;DR

This paper investigates the algebraic structure of Rabinowitz--Floer homology on contact manifolds with periodic Reeb flows, revealing it as a Laurent polynomial module with a pair-of-pants product structure.

Contribution

It establishes that RFH* forms a Laurent polynomial module over Z_2[s,s^{-1}] with a pair-of-pants product, under periodic Reeb flow conditions.

Findings

RFH* is a module over Laurent polynomials Z_2[s,s^{-1}]

The module is often free and finitely generated

The module structure is given by the pair-of-pants product

Abstract

In this paper, we explore the structure of Rabinowitz--Floer homology $RFH_*$ on contact manifolds whose Reeb flow is periodic (and which satisfy an index condition such that $RFH_*$ is independent of the filling). The main result is that $RFH_*$ is a module over the Laurent polynomials $\mathbb{Z}_2[s,s^{-1}]$, where $s$ is the homology class generated by a principal Reeb orbit and the module structure is given by the pair-of-pants product. In most cases, this module is free and finitely generated.